Okay, let’s take the singular value decomposition and do something really neat with it. Specifically, we’ll start with an endomorphism and we’ll write down its singular value decomposition
where and are in the unitary (orthogonal) group of . So, as it turns out, is also unitary. And is positive-semidefinite (since is). And, of course, since , we can write
That is, any endomorphism can be written as the product of a unitary transformation and a positive-semidefinite one.
Remember that unitary transformations are like unit complex numbers, while positive-semidefinite transformations are like nonnegative real numbers. And so this “polar decomposition” is like the polar form of a complex number, where we write a complex number as the product of a unit complex number and a nonnegative real number.
We can recover the analogy like we did before, by taking determinants. We find
since the determinant of a unitary transformation is a unit complex number, and the determinant of a positive-semidefinite transformation is a nonnegative real number. If is nonsingular, so is actually positive-definite, then will be strictly positive, so the determinant of will be nonzero.
We could also define , so . This is the left polar decomposition (writing the positive-definite part on the left), where the previous form is the right polar decomposition